However, for realistic model, the bond wires are also considered for industry benchmarks (industry1-industry6) during parasitic extraction, and are modeled as parasitic resistances.
Table 4.3 lists details on all benchmark circuits considered for performance assessment.
Table 4.3:Power distribution benchmark statistics
Benchmarks Benchmark Statistics1 IR drop profile (V) Analysis (KLU Solver)
#n #i #v #r Max. Min. Avg. ϑ(%)2 Time (min:sec)
ibmpg2 127236 37926 330 208325 0.6160 0.00012 0.2860 47.36 00:06 ibmpg4 953581 276976 962 1560645 0.3860 0.00011 0.0051 36.18 00:10 ibmpg5 1079308 540800 539087 1076848 0.0520 0.00022 0.0127 35.89 00:21 ibmpg6 1670492 761484 836239 1649002 0.2170 0.00021 0.0227 42.06 00:35 industry1 250K 248974 1029 499000 1.6721 0.00020 0.1918 32.93 00:06 industry2 1M 998943 1059 1998000 1.0288 0.00020 0.5310 47.18 00:16 industry3 4M 3996980 3020 7996000 1.5315 0.00013 0.5721 53.71 02:02 industry4 9M 8994790 5211 17994000 1.6511 0.00021 0.5140 68.56 08.04 industry5 16M 15991481 8523 31992000 1.7013 0.00020 0.3856 76.57 22.39 industry6 25M 24987989 12013 49990000 1.7982 0.00018 0.2081 89.67 54.06
1 #ndenotes number of nodes,#idenotes number of current sources,#vdenotes number of voltage sources,#rdenotes number of resistances.
2 ϑdenotes percentage of affected nodes (above threshold) before optimization.
4.5 Design Optimization Process
4.5.1 Preprocessing step
Before optimization, steady-state analysis is performed to generate the IR drop profile of the PDN benchmarks as presented in [131]. As the coefficient matrix (ArGGATrG) represent- ing the conductance network (steady state) is symmetric and positive definite [114], it can be simulated efficiently using existing power distribution network analyzers. During imple- mentation, we employ KLU [132] for fast steady state analysis of power distribution network.
The maximum, minimum and average IR drop values all PDN benchmarks are reported in Table 4.3 along with computational time involved in analyzing PDN benchmarks. It can be observed that with increase in number of nodes of power distribution networks, computational cost in analyzing these networks also increases, which affects the optimization process. Once analysis is performed, power distribution network is subjected to the optimization process for evaluating minimum IR drop and wire area subjected to various constraints.
4.5.2 Voltage drop at each node and minimization using modified river formation dynamics algorithm
Considering the power distribution network as a linear system as described in (4.1), the node potentialsvnare evaluated by performing either steady state analysis (on resistive net- work) or transient analysis (on RC/RLC-network) on the entire power distribution network before beginning the optimization process. This evaluation gives us a complete IR drop pro- file of the entire network as shown in Figure 1.4(a). For each node x as shown in Figure 4.1(c), the voltage drop corresponding to the metal widths of branches (e.g.,ρlxygxy) or im- pulse current excitations (considered during transient analysis) at all the nodes is affected by the voltage drops at neighbor nodes corresponding to their connected metal widths and impulse current excitations. In fact, a node having IR drop affects its neighbor and, subse- quently, affects a small region as shown in Figure 1.4(a). In view of this, the problem of IR drop at any nodexcan also be represented as (4.26),
ϑx =Vdd−vx =Vdd−
∑deg(x) i=1 givi
∑deg(x) i=1 gi
+ ix
∑deg(x) i=1 gi
. (4.26)
To minimize ϑx described in (4.26) subject to constraints in (4.5), deg(x) + 1 number of decision variables are considered which numerically constitute branch conductances (gi,i=1 to deg(x)) and a current sink (ix or Isink). The proposed MRFD algorithm is designed to minimize IR drop of power distribution network as described in Algorithm 13. For each node, the neighbor branch conductances and current sink value are provided as a bound, i.e., being evaluated through steady state analysis (or transient analysis for RC/RLC power distribution network). The process of minimization starts by considering the decision variables as multiple volumes of sediment rates per unit time per unit width (qs), which are varied in each time step using (4.13) guided by constriction coefficients (w1, w2, w3) and probabilistic seed value (ε) as described in section 4.3.8. The constriction coefficients are generated within (1,2) as described in section 4.3.8) during evaluation. However, ε is evaluated for each decision variable by following the procedure described in Algorithm 12. Longitudinal slope (Sl) is
4.5 Design Optimization Process
Algorithm 13:IR drop minimization procedure
input :A power grid network of sizeN.
output:Optimal IR drop at each nodex∈N.
Data:ArGGATrG,ArJiJ: Conductance and current sink matrices defined in section 4.2 whilei < numExpdo
foreach nodex∈Ndo ifVdd−vx< Vththen
Minimizeϑxin (4.26) subject to constraints in (4.5) Evaluate optimal IR drop (ϑx) using MRFD
/* update current node specification */
Update optimal potential (vx), branch conductances and current sink x←−x+ 1
end end i←−i+ 1 end
evaluated from altitude values (h1, h2 as described in section 4.3.6) and distance between two altitude points (x1 − x2). During evaluation, the current potential at nodes represent different altitude and the conductances between neighbors correspond to distance. On the other hand, transverse slope (Sn) is evaluated using (4.11). Initial random generated values of decision variable (conductances or current sink) is assigned to qs and, qn and β are kept fixed at1and1.4, respectively, for evaluation of Sn(positive slope). Further, us andun are assigned to potentials of current node and the neighboring nodes, respectively. In this manner, the slopes and corresponding probability values of all nodes are evaluated and considered during estimation ofε. Such implementation administers the evaluation of IR drop in optimal direction. Once the minimization is complete at a node, the corresponding node potential, current sink value and branch conductances are updated, and the process is shifted to another affected node (IR drop node) for minimization. This process of minimization is performed on the affected nodes for25times independently and the median of optimal IR drop is considered as the global optimum solution.
4.5.3 Minimizing metal area using MRFD method
Similar to the implementation of RFD scheme, MRFD method is employed to minimize metal area of power distribution network. The IR drop profile is evaluated before start of op- timization process. Further, instead of analyzing a single branch of metal wire at time, several
cells are analyzed for the optimization of wire width. Wire widths (e.g.,wxy) are considered as decision variables (qs), which are updated in each iteration by following (4.21). Similar to the minimization of IR drop, the values constriction coefficients (w1, w2, w3) are set within (1,2) and the probabilistic seed value (ε) is evaluated using Algorithm 12. As similar constraints affect both IR drop and metal area, both random boundary handling strategy and constrained- dominance principle are employed for handling the constraints. A number of iterations are carried out on a single cell to evaluate the minimum width of each branch of the cell. Once the minimization is complete, corresponding cell is considered for the evaluation. The branches which are already being evaluated for minimum width (i.e., the overlapping branch between two cells) are not considered for further evaluation. The process of minimization at a single cell is performed for 25times independently and the median of optimal width is considered as global optimum solution. In this way, wire widths corresponding to the affected nodes are optimized and total wire area of the power distribution network is evaluated at the end.
As wire widths are evaluated in the optimal direction, after the evaluation, network area is also reduced in an optimal way. Such evaluation is important for designers as it serves as a guideline for power distribution network optimization.